Vector Space Subspace Proof Suppose that W is a subspace of a finite-dimensional vector space V . Prove
that W = V if and only if dim W = dim V.
This is what I did: 
Suppose dim W =dim V$\iff|$basis of $W|=|$basis of $V|$ and since W $ \subseteq V \implies W=V$.
For the converse, if $W=V $ then basis of W = basis of V. So $ |$basis of $W|=|$basis of $V|$ and hence dim W =dim V.
Is this enough? Is it right in the first place? Please help.
 A: Your proof of the converse looks fine to me - though I'd probably write it as "if $S$ is a basis for $W$ it is also a basis for $V$, hence the dimension of each equals $|S|$ and are thus equal to each other" since your current wording makes it kind of look like bases are unique, whereas we really are choosing a particular basis and saying something about it. But the spirit of it is correct.
I don't follow your first proof, though. The property that $W\subseteq V$ and that $|\text{basis of $W$}|=|\text{basis of $V$}|$ are both true, but it's not clear what reasoning combines them into $V=W$. What you might be getting at is that any basis $S$ for $W$ would be a linearly independent set in $V$. However, given that $V$ has dimension $|S|=\dim W=\dim V$, it follows that any linearly independent set of size $|S|$ is a basis for $V$. This means that any basis for $W$ is a basis for $V$. Then, it follows that $W$ and $V$ are both described as the span of $S$ (i.e. the set of linear combinations of its elements), and are hence equal.
